Error analysis of a highly efficient and accurate temporal multiscale method for a fractional differential system

This work is concerned with a nonlinear coupled system of fractional ordinary differential equations (FODEs) with multiple scales in time. The micro system is a nonlinear differential equation with a periodic applied force and the macro system is a fractional differential equation. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled fractional problem at a lower computational cost, which calls for us to develop an effective analysis and computation framework. We derive a periodic auxiliary system to approximate the original system and give error estimates. Then we propose a numerical scheme based on the auxiliary problem and analyze the discrete error. An efficient multiscale algorithm that is also applicable to 2-D and 3-D problems is designed. The results of numerical experiments demonstrate the accuracy and computational efficiency of the proposed multiscale method. It is observed that, the computational time is significantly reduced and the multiscale method performs very well in comparison to the fully resolved simulation. Furthermore, as a particular example of applications, we consider a simplified model of the atherosclerotic plaque growth problem, and simply evaluate the effect of fractional parameter and damping coefficient on plaque growth and discuss the physical implication.